Why the Hausdorff Axiom Was Not Part of the Original Definition of Topological Spaces
Topological spaces, a fundamental concept in mathematics, come in various flavors, with the Hausdorff axiom (T2 axiom) being just one of the many conditions that can be imposed on them. Despite the prevalence of the definition of topological spaces as a set with a family of open sets, the Hausdorff axiom is not always included in this definition, as testified by my topology professor and countless examples of spaces that violate this condition.
Historical Context and Evolution of Topological Space Definitions
The prominence of the concept of a topological space as we know it today is at least partly due to the historical winnowing process of related ideas. Certain axioms, such as the Hausdorff condition, emerged as particularly significant, while others, like the uniform space concept, did not gain the same level of popularity or recognition among mathematicians.
For instance, when I was a graduate student, the concept of a uniform space was a topic that received less attention, with the term 'uniform space' appearing only occasionally in coursework. In contrast, the term 'Hausdorff topological space' is far more frequently used and recognized, likely due to the importance of Hausdorff spaces in topology and the broader mathematical community.
The Hausdorff Axiom: Not a Mandatory Condition
It is important to understand that not all topological spaces need to be Hausdorff. The T-axioms, including the Hausdorff axiom, were developed to classify and study different types of topological spaces. While Hausdorff spaces are relatively ordinary in comparison to more exotic spaces, they remain a significant part of the study of topology.
The Emergence of Other Names and Categories
The emergence of other names, such as 'metric space' and 'Polish space,' reflects a historical development where certain categories of topological spaces gained recognition and their own names. This naming process is partly a result of the historical context and the evolution of mathematical thought.
The Polish space, for example, is a topological space that is metrizable and has a countable dense subset. While 'Polish space' is less frequently used than 'metric space,' both have become important categories within the subfield of topology.
Equivalent Axioms and Names
Just as the transition from ring to commutative ring reflects a naming evolution, the naming of topological spaces reflects similar shifts. In abstract algebra, older texts may have used 'ring' to denote what is now commonly referred to as a commutative ring. Similarly, the term 'ring' without the commutativity condition is now more prevalent, but authors may still specify commutativity when needed.
The Importance of Non-Hausdorff Spaces
While Hausdorff spaces are well-studied, non-Hausdorff spaces also play a crucial role in various branches of mathematics, particularly in logic and algebraic geometry. The Zariski topology, a common topology in algebraic geometry, is typically non-Hausdorff. In this topology, the generic point belongs to every open set, making it a prime example of a non-Hausdorff space.
Non-Hausdorff spaces are also connected to partial orderings. Many non-Hausdorff topologies have been shown to be related to order structures, providing a rich area for further study and exploration.
Even when starting with a Hausdorff space, quotienting by an equivalence relation can result in a non-Hausdorff space. This is particularly relevant in the study of group actions on topological spaces. When considering the quotient topology, the closure of an orbit containing a point may intersect other orbits, leading to a non-Hausdorff quotient.
Conclusion
The omission of the Hausdorff axiom from the original definition of topological spaces is not a mere oversight but a conscious choice based on the historical development and the evolution of mathematical thought. While Hausdorff spaces are indeed special and important, the broader category of topological spaces, including non-Hausdorff spaces, offers a rich and diverse landscape for exploration and application.